Summary: Hilbert's -Terms in
Automated Theorem Proving
Martin Giese and Wolfgang Ahrendt
Institut f¨ur Logik, Komplexit¨at und Deduktionssysteme,
Universt¨at Karlsruhe, Germany
{giese|ahrendt}@ira.uka.de
Abstract. -terms, introduced by David Hilbert [8], have the form x.,
where x is a variable and is a formula. Their syntactical structure is
thus similar to that of a quantified formulae, but they are terms, denoting
`an element for which holds, if there is any'.
The topic of this paper is an investigation into the possibilities and lim-
its of using -terms for automated theorem proving. We discuss the re-
lationship between -terms and Skolem terms (which both can be used
alternatively for the purpose of -quantifier elimination), in particular
with respect to efficiency and intuition. We also discuss the consequences
of allowing -terms in theorems (and cuts). This leads to a distinction
between (essentially two) semantics and corresponding calculi, one en-
abling efficient automated proof search, and the other one requiring hu-
man guidance but enabling a very intuitive (i.e. semantic) treatment of
-terms. We give a theoretical foundation of the usage of both variants in